Determine n if C(2n,3):C(n,3)=12:1

To determine the value of n in the equation C(2n, 3) : C(n, 3) = 12 : 1, we can follow these steps:

Step 1: Understand the equation The equation compares two combinations, C(2n, 3) and C(n, 3), in a ratio of 12:1. We need to find the value of n that satisfies this ratio.

Step 2: Calculate the combinations Using the formula for combinations, we can calculate the values of C(2n, 3) and C(n, 3).

C(2n, 3) = (2n)! / (3!(2n-3)!) C(n, 3) = n! / (3!(n-3)!)

Step 3: Simplify the equation Now we can substitute the calculated combinations into the equation and simplify it.

(C(2n, 3)) / (C(n, 3)) = 12 / 1 ((2n)! / (3!(2n-3)!)) / (n! / (3!(n-3)!)) = 12

Step 4: Simplify the equation further To simplify the equation, we can cancel out common terms in the numerator and denominator.

((2n)(2n-1)(2n-2)) / (3(2n-3)(2n-4)) = 12

Step 5: Solve for n Now we can solve the equation for n. Cross-multiply and simplify.

((2n)(2n-1)(2n-2)) = 12(3(2n-3)(2n-4)) (2n)(2n-1)(2n-2) = 12(6n-18)(2n-4) (2n)(2n-1)(2n-2) = 72(2n-3)(2n-4)

Step 6: Solve the equation At this point, we have a cubic equation that we need to solve for n. We can use algebraic methods or numerical methods to find the value of n that satisfies the equation.

Once we find the value of n, we can substitute it back into the original equation to verify if it indeed satisfies the ratio of 12:1.